The Fundamental Theorem on Symmetric Polynomials: History's First Whiff of Galois Theory

Blum-Smith, Ben; Coskey, Samuel

doi:10.4169/college.math.j.48.1.18

Cited by 12 publications

(11 citation statements)

References 7 publications

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“…Let us now explore the concept of approximation of symmetric functions by symmetric polynomials. For more details, we refer the reader to Davidson and Donsig ( 2009 ), Blum-Smith and Coskey ( 2017 ). In the sequel, we will denote the symmetric group over the set {1, …, n } as S n .…”

Section: Approximating Symmetric Functions With Symmetric Polynomialsmentioning

confidence: 99%

On the Construction of Group Equivariant Non-Expansive Operators via Permutants and Symmetric Functions

Conti

Frosini

Quercioli

2022

Front. Artif. Intell.

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Group Equivariant Operators (GEOs) are a fundamental tool in the research on neural networks, since they make available a new kind of geometric knowledge engineering for deep learning, which can exploit symmetries in artificial intelligence and reduce the number of parameters required in the learning process. In this paper we introduce a new method to build non-linear GEOs and non-linear Group Equivariant Non-Expansive Operators (GENEOs), based on the concepts of symmetric function and permutant. This method is particularly interesting because of the good theoretical properties of GENEOs and the ease of use of permutants to build equivariant operators, compared to the direct use of the equivariance groups we are interested in. In our paper, we prove that the technique we propose works for any symmetric function, and benefits from the approximability of continuous symmetric functions by symmetric polynomials. A possible use in Topological Data Analysis of the GENEOs obtained by this new method is illustrated.

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Section: Approximating Symmetric Functions With Symmetric Polynomialsmentioning

confidence: 99%

On the Construction of Group Equivariant Non-Expansive Operators via Permutants and Symmetric Functions

Conti

Frosini

Quercioli

2022

Front. Artif. Intell.

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“…Meanwhile, the invariance of ab under (8) is a special case of the fact that ab = (e iθ a)(e −iθ b) for any θ, and this latter invariance property is behind all the cancellation in Price's key calculation (5). 3 Second, the reason for the utility of shifting n (z) upward by a n + b n -arriving at P n (z), which satisfies a nice recursion-remains a mystery in [21,22]. We view Proposition 1 as the beginning of a resolution: n (z) was something natural minus a n + b n in the first place.…”

Section: Symmetric Polynomials and Cardano's Depressed Cubicmentioning

confidence: 99%

“…The classic proof is found in an 1816 paper of Gauss[10, paragraphs 3-5]. For an in-depth discussion of this proof, see[3]. Gauss's paper also appears to contain the first modern statement of the theorem; all prior articulations formulated it as a statement about roots of a polynomial, implicitly assuming the existence of a splitting field.…”

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Chords of an Ellipse, Lucas Polynomials, and Cubic Equations

Blum-Smith¹,

Wood

2020

The American Mathematical Monthly

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A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse, generalizing a classic problem about circles. We give a brief history of the circle problem, an account of Price's ellipse proof, and a reorganized proof, with some new ideas, designed to situate the result within a dense web of connections to classical mathematics. It is inspired by Cardano's solution of the cubic equation and Newton's theorem on power sums, and yields an interpretation of generalized Lucas polynomials in terms of the theory of symmetric polynomials. We also develop additional connections that surface along the way; e.g., we give a parallel interpretation of generalized Fibonacci polynomials, and we show that Cardano's method can be used to write down the roots of the Lucas polynomials.

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“…). The underlying set of N 2=4 is {[0],[1],[2],[3]}, and we have[0] ≤ [1] ≤ [2] ≤ [3] ≤ [2]. The uc-semiring N 2=4 is Frobenius, but not upper-bound or symhomomorphic.Is it fully elementary?…”

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confidence: 99%

Symmetric polynomials in upper-bound semirings

Kališnik

Lešnik

2021

Journal of Symbolic Computation

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The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is fully elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The result does not extend directly to polynomials over semirings, but we do have analogous results for some special semirings, for example, the tropical, extended and supertropical semirings. These all fall into a larger class of upper-bound semirings. In this paper we extend the known results and give a complete characterization of fully elementary upper-bound semirings. We further improve this characterization statement in the case of linearly ordered upper-bound semirings.

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The Fundamental Theorem on Symmetric Polynomials: History's First Whiff of Galois Theory

Cited by 12 publications

References 7 publications

On the Construction of Group Equivariant Non-Expansive Operators via Permutants and Symmetric Functions

On the Construction of Group Equivariant Non-Expansive Operators via Permutants and Symmetric Functions

Chords of an Ellipse, Lucas Polynomials, and Cubic Equations

Symmetric polynomials in upper-bound semirings

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